A class of graphs is structurally nowhere dense if it can be constructed from a nowhere dense class by a first-order transduction. Structurally nowhere dense classes vastly generalize nowhere dense classes and constitute important examples of monadically stable classes. We show that the first-order model checking problem is fixed-parameter tractable on every structurally nowhere dense class of graphs. Our result builds on a recently developed game-theoretic characterization of monadically stable graph classes. As a second key ingredient of independent interest, we provide a polynomial-time algorithm for approximating weak neighborhood covers (on general graphs). We combine the two tools into a recursive locality-based model checking algorithm. This algorithm is efficient on every monadically stable graph class admitting flip-closed sparse weak neighborhood covers, where flip-closure is a mild additional assumption. Thereby, establishing efficient first-order model checking on monadically stable classes is reduced to proving the existence of flip-closed sparse weak neighborhood covers on these classes - a purely combinatorial problem. We complete the picture by proving the existence of the desired covers for structurally nowhere dense classes: we show that every structurally nowhere dense class can be sparsified by contracting local sets of vertices, enabling us to lift the existence of covers from sparse classes.

翻译：图类称为结构无处稠密类，若它可由一个无处稠密类通过一阶转换构造而成。结构无处稠密类极大地推广了无处稠密类，构成了单一定稳定图类的重要范例。我们证明：在每个结构无处稠密图类上，一阶模型检测问题是固定参数可解的。该结果基于近期发展的单一定稳定图类的博弈论刻画。作为另一具有独立意义的关键要素，我们提出一种多项式时间算法，用于逼近（一般图上的）弱邻域覆盖。我们将这两种工具结合，形成一种基于递归局部性的模型检测算法。该算法在每一类允许翻转闭稀疏弱邻域覆盖的单一定稳定图类上高效运行，其中翻转闭是一个温和的附加假设。由此，在单一定稳定图类上建立高效的一阶模型检测可归结为证明这些类上存在翻转闭稀疏弱邻域覆盖——这是一个纯组合问题。我们通过证明结构无处稠密类上存在所需覆盖来完善此图景：我们表明，每个结构无处稠密类可通过收缩局部顶点集进行稀疏化，从而将覆盖的存在性从稀疏类提升至此类。